Chapter A.1, Problem 64E

Solution

Simplify the exponential expression.

  $6{\left(\frac{x}{y}\right)}^{7/6}$ .

Given:

The given expression is $\left(\frac{2x^{1/2}}{y^{2/3}}\right)\left(\frac{3x^{2/3}}{y^{1/2}}\right)$ .

Concept Used:

• Exponential expressions are just a way to write powers in short form. The exponent indicates the number of times the base is used as a factor.

Calculation:

The given expression is $\left(\frac{2x^{1/2}}{y^{2/3}}\right)\left(\frac{3x^{2/3}}{y^{1/2}}\right)$ .

  $\begin{array}{c}\left(\frac{2x^{1/2}}{y^{2/3}}\right)\left(\frac{3x^{2/3}}{y^{1/2}}\right)=\frac{6x^{1/2}{x}^{2/3}}{y^{2/3}{y}^{1/2}}\\ =\frac{6x^{\frac{1}{2}+\frac{2}{3}}}{y^{\frac{2}{3}+\frac{1}{2}}}\\ =\frac{6x^{\frac{3+4}{6}}}{y^{\frac{4+3}{6}}}\\ =\frac{6x^{\frac{7}{6}}}{y^{\frac{7}{6}}}\\ =6{\left(\frac{x}{y}\right)}^{7/6}\end{array}$

Thus after simplifying the given expression $\left(\frac{2x^{1/2}}{y^{2/3}}\right)\left(\frac{3x^{2/3}}{y^{1/2}}\right)$ is $6{\left(\frac{x}{y}\right)}^{7/6}$ .